>In the condition Progeria children born with this congenital disorder begin to assume the appearence and disabilities of old age typically well before they reach ten years of age.<
The primary defect in these children is not shortened telomeres. It's a mutation in a DNA helicase that is involved in homologous DNA repair and, probably, in nonhomologous end-joining pathways. The gene (WRN) may be involved in resolving something called the Holliday junction, a 4-stranded DNA intermediate in the homologous DNA repair pathway. If the Holiday junction is not resolved, it can lead to multiple DNA breaks, which of course may lead to all sorts of problems if improperly repaired or left unrepaired.
In short, aging--whether it be 'natural', or accelerated by environmental insults (eg, UV radiation, ionizing radiation, smoking, caffeine, etc) is probably caused primarily by unrepaired or improperly repaired DNA damage.
Shortened telomeres are a secondary consequence of Werner's syndrome. I don't think this is the primary defect.